Why didn't Hermes drop the supply probe on Mars?

They didn't have anything that could land on Mars. There was no way for them to drop off supplies to the surface.

The resupply probe they got from their Earth fly-by. The original plan was they were going to build it to crash land on Mars, but since the Hermes crew tipped their hand, they didn't have time to make it crash proof because now it just needed to dock with Hermes near the Earth.

The resupply probe attached to one of the airlocks.

When it arrives at Mars, the probe isn't still there. So they must have dumped it after getting the supplies to reduce their mass to make their journey to Mars a little lighter / faster.

Here you can clearly see the two airlocks with no probe attached to either one (it would have been the lower one in this picture).

Additionally, had they just resupplied Watney, it would have meant him having to spend another two years or more waiting for Hermes to return to Earth and the bring back the Aries 4 crew.

This was addressed more fully in the film's source novel. In short, the bigwigs at NASA wanted to go with your idea, dropping the supply payload near Watney and hoping that he's able to survive until the next ship gets there on schedule.

“I see,” Teddy said. “So we have two options on the table: Send Watney enough food to last until Ares 4, or send Hermes back to get him right now. Both plans require the Taiyang Shen, so we can only do one.”
“Yes,” Venkat said. “We’ll have to pick one.”

...

“Consider degree of risk, Teddy,” Venkat said. “Mitch is right. The crash-lander is high-risk. It could miss Mars, it could reenter wrong and burn up, it could crash too hard and destroy the food…We estimate a thirty percent chance of success.”
“A near-Earth rendezvous with Hermes is more doable?” Teddy asked. “Much more doable,” Venkat confirmed. “With sub-second transmission delays, we can control the probe directly from Earth rather than rely on automated systems. When the time comes to dock, Major Martinez can pilot it remotely from Hermes with no transmission delay at all. And Hermes has a human crew, able to overcome any hiccups that may happen. And we don’t have to do a reentry; the supplies don’t have to survive a three-hundred-meters-per-second impact.”

NASA Administrator Teddy Sanders personally signed off on the "Iris 2 plan" (the supply drop option).

“It wasn’t an easy decision,” Teddy said to the assembled elite. “But I’ve decided to go with Iris 2. No Rich Purnell Maneuver.”
Mitch slammed his fist on the table.
“We’ll do all we can to make it work,” Bruce said.
“If it’s not too much to ask,” Venkat began, “what made up your mind?”
Teddy sighed. “It’s a matter of risk,” he said. “Iris 2 only risks one life. Rich Purnell risks all six of them. I know Rich Purnell is more likely to work, but I don’t think it’s six times more likely.”

But when it came down to it, the crew made the decision for them. They were willing to risk their lives to save Watney.

Lewis leaned forward. “There was more in the message,” she began. “We’d have to pick up a supply near Earth, and he’d have to get to Ares 4’s MAV.”
“Why all the cloak and dagger?” Beck asked.
“According to the message,” Lewis explained, “NASA rejected the idea. They’d rather take a big risk on Watney than a small risk on all of us. Whoever snuck it into Vogel’s e-mail obviously disagreed.”
“So,” Martinez said, “we’re talking about going directly against NASA’s decision?”
“Yes,” Lewis confirmed, “that’s exactly what we’re talking about. If we go through with the maneuver, they’ll have to send the supply ship or we’ll die. We have the opportunity to force their hand.”
“Are we going to do it?” Johanssen asked.
They all looked to Lewis.

...

“If we do this,” Vogel said, “it would be over one thousand days of space. This is enough space for a life. I do not need to return.”
“Sounds like Vogel’s in,” Martinez grinned. “Me, too, obviously.”
“Let’s do it,” Beck said.
“If you think it’ll work,” Johanssen said to Lewis, “I trust you.”

They can't just drop the supply probe on the planet. There isn't enough delta-V. So the Hermes is swinging past Earth and already picking up speed (they're not in orbit). That's why they need such a huge booster just to get it up to speed with the Hermes. The Hermes is still accelerating and flies past Mars at an incredible speed (actually 5.8 km/s according to this answer).

That's why they had to lighten the MAV to get Watney to rendezvous with the Hermes. The MAV was designed to get you to low Mars orbit, which was "only" 4.1 km/s. You'd need to impart the full 5.8 km/s to get the supply probe to slow down enough to land (even crash landing) on Mars. If you had the probe hit Mars at 5.8 km/s there won't be anything left but molten rock mixed with melted protein bars. You still have to decelerate to a speed where the payload will be relatively intact (I'm thinking less than 0.1 km/s).

Mars' atmosphere isn't thick enough to slow down all the way, even with parachutes. Those supplies had to supply the entire Hermes mission, which means it was heavier than the probe you could have sent laden with food to Watney directly, which is why there would be enough weight left over to include a descent module to land it on Mars.

That brings us back to the choice in the movie: they can either try sending another probe directly to Watney so he can wait until Ares 4, or they can supply Hermes with enough to go get him, but they can't do both.

The real question is, Why would they risk Watney's life for another two years? ...when they've already chosen option (b).

Option (a): send the probe on a fly-by trajectory towards Mars, use aerobraking and what little fuel it has to be captured, and eventually impact the surface without 1) destroying its cargo, and 2) landing more than 60 km away from Watney's position, and hope that Mark can survive another two years without 3) another airlock tearing open, 4) his water-reclaimer breaking (= dead), 5) his oxygenator breaking (= dead), 6) his hab, designed for a 31-day mission and already greatly overused, ripping open in some other location besides the airlocks (= most likely dead), and finally 7) Mark losing sanity from isolation for 2 additional years.

Remember at this point, Mark has accidentally fried Pathfinder so he once again has no way to talk to NASA. He modified the drill he was using to cut holes in the rover from using batteries to running off hab power from a cable, and leaned it up against Pathfinder with exposed wires... oops. Hab power has 6 amps, Pathfinder was only built for 0.6 amps. At this point, he had to get to the Ares IV MAV just to make contact again. In the book, Mark's great sense of humor is what keeps him sane most of the time, but two years is a serious test on anyone's sanity.

Option (b): send the Hermes on a fly-by trajectory to Earth for gravity assist to a fly-by trajectory to Mars, hoping that 1) the resupply from the Taiyang Shen is successful, 2) Mark can modify the MAV to be light enough to reach Mars' escape velocity and rendezvous with the Hermes, and 3) the Hermes can survive deep space for another 2 years past its maintenance window.

So essentially, you question is "why, when the Hermes was already on its way to Mars, did it not just drop off the care package and leave Mark for another 2 years?" "This would've been safer for Watney and would've saved the Ares IV MAV"

Let's call this option (c), and you will see very soon here that all we're doing is stacking the worst risks from both options (High probability of killing 1 person vs. low probability of killing 6 people) with the only real payoff being: We get to keep the Ares IV MAV.

Option (c): send the Hermes on a fly-by trajectory to Earth for gravity assist to a fly-by trajectory to Mars, hoping that b-1) the resupply from the Taiyang Shen is successful, b-3) the Hermes can survive deep space for another 2 years past its maintenance window, release the probe on approach at escape velocity towards Mars, use aerobraking and what little fuel it has to be captured, and eventually impact the surface without a-1) destroying its cargo, and a-2) landing more than 60 km away from Watney's position, and hope that Mark can survive another two years without a-3) another airlock tearing open, a-4) his water-reclaimer breaking (= dead), a-5) his oxygenator breaking (= dead), a-6) his hab, designed for a 31-day mission and already greatly overused, ripping open in some other location besides the airlocks (= most likely dead), and finally a-7) Mark losing sanity from isolation for 2 additional years.